If you leave Earth heading for the sun, you should get a gravity assist on flyby.
When you travel in space, strictly speaking all masses attract each other (n-body-problem). However most time, there is one central body, whose gravity is dominant and exceeds other forces by several orders of magnitude. For traveling between planets in our solar system, this is clearly the sun.
A gravity-assist or slingshot means you approach another body, so its gravity exceeds the suns for a short time, the dominant central body changes. If you look at this in a sun reference frame, this bends your trajectory. The trick of this maneuver is, that in the assisting planets frame, you do not loose or gain energy, but in the suns frame you do, because the frames a moving against each other.
Can our sun itself assist you? No, it can't, because you are already under it's influence, the trick with changing the dominant body does not work here (unless you come from outside our solar system).
It happens that also other assists (except luna) will not help, because reaching the assisting body needs more energy than reaching mars directly. A Hohmann-transfer is the cheapest way to get there.
At least in my mind, the travel time to Mars would be the same no matter where the two plants are.
Now Hohmann will arrange for getting to mars' orbit, but theres nothing to be found there unless mars happens to be right at this point when you arrive. Braking there and waiting until mars arrives does also not work, you need to keep your speed relative to the sun in order to stay there. So you and mars would be traveling at the same speed and never meet. Starting at the right time is essential.
The same amount of fuel needed to get out to Mars, must be enough to get to the Mars via the Sun.
No, Walter Hohmann showed in his work The attainability of heavenly bodies that this is not the case. In order to approach sun, you need to slow down in suns reference frame to lower centrifugal forces. To get to mars, you need to accelerate, because it is farther out.
Is the fastest way...?
If you can afford the thrust all the time, a direct intercept to your target is always fastest. The pointing of your thrusters needs to be tilted somewhat to counteract the Coriolis force. Low consumption thrusters strong enough to escape from earth are not available (as of 2016).
With limited propellants, mission duration will be shortest if you use them in the beginning and at the end, which again means a Hohmann trajectory.
Earliest arrival after a given point in time is different, because you will not be able to delay your start until a good launch opportunity. For earliest arrival, you may need a completely different trajectory. Gravity-assists may be beneficial in this case to leave the ecliptic.
